# Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?

I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on nLab) of the projective level model structure on symmetric spectra, which has level equivalences as weak equivalences.

Almost all ingredients are there in the article. All I have left to show is that the injective Omega-spectra are indeed the S-local objects, where S is the class of stable equivalences. By defintion any map in S induces a weak equivalence of simplicial hom-sets $Map_{Sp^\Sigma}(f,E)$ for E a injective Omega-spectrum. Conversely, lemma 3.1.5 and example 3.1.10 conspire to tell you that if a symmetric spectrum is S-local and injective, it is an Omega-spectrum. So, what remains: is any S-local symmetric spectrum injective?

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You said "...where S is the set of level equivalences." Do you mean S is the set of stable equivalences? –  Tyler Lawson Dec 9 '09 at 13:14
Yes, in fact example 3.1.10 doesn't work when you define S to be the class of level equivalences, because the lambda used there is not a pi_*-isomorphism, but all level equivalences are. –  skupers Dec 9 '09 at 13:28

You have to realize it has been a long time since we wrote that paper. But I'll give it my best shot.

I think we intentionally chose the injective Omega-spectra because they are "extra fibrant", so to speak. That is, I think S-local spectra don't have to be injective, just Omega-spectra.

The injective Omega-spectra should be the fibrant objects in a different model structure. There should be an injective level structure, which I guess we did not construct, where the cofibrations are monomorphisms and the weak equivalences are level equivalences. The fibrant objects would then be the injective spectra. The injective Omega-spectra are then the fibrant objects in the left Bousfield localization of this category with respect to the stable equivalences.

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